continuous


Let X and Y be topological spacesMathworldPlanetmath. A function f:XY is continuousMathworldPlanetmath if, for every open set UY, the inverse image f-1(U) is an open subset of X.

In the case where X and Y are metric spaces (e.g. Euclidean spaceMathworldPlanetmath, or the space of real numbers), a function f:XY is continuous if and only if for every xX and every real number ϵ>0, there exists a real number δ>0 such that whenever a point zX has distance less than δ to x, the point f(z)Y has distance less than ϵ to f(x).

Continuity at a point

A related notion is that of local continuity, or continuity at a point (as opposed to the whole space X at once). When X and Y are topological spaces, we say f is continuous at a point xX if, for every open subset VY containing f(x), there is an open subset UX containing x whose image f(U) is contained in V. Of course, the function f:XY is continuous in the first sense if and only if f is continuous at every point xX in the second sense (for students who haven’t seen this before, proving it is a worthwhile exercise).

In the common case where X and Y are metric spaces (e.g., Euclidean spaces), a function f is continuous at xX if and only if for every real number ϵ>0, there exists a real number δ>0 satisfying the property that dY(f(x),f(z))<ϵ for all zX with dX(x,z)<δ. Alternatively, the function f is continuous at aX if and only if the limit of f(x) as xa satisfies the equation

limxaf(x)=f(a).
Title continuous
Canonical name Continuous
Date of creation 2013-03-22 11:51:55
Last modified on 2013-03-22 11:51:55
Owner djao (24)
Last modified by djao (24)
Numerical id 12
Author djao (24)
Entry type Definition
Classification msc 26A15
Classification msc 54C05
Classification msc 81-00
Classification msc 82-00
Classification msc 83-00
Classification msc 46L05
Synonym continuous function
Synonym continuous map
Synonym continuous mapping
Related topic Limit
Defines continuous at